发表机构
Katholische Universität Eichstätt-Ingolstadt（因戈尔施塔特天主教大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文引入辛指标间隙参数，对给定格上 Gabor 窗的衰减与光滑性给出分类，证明 Schwartz 类窗存在条件，并统一了经典与融合 Balian-Low 定理。
AI 中文摘要
格 Gabor 框架由窗函数及其在格元素上的时频平移组成。本文研究在给定时间-频率/相空间格下，Gabor 框架窗函数可达到的衰减性与光滑性，并给出了除下述边界情形之外的分类。本文的核心是引入一个整数格参数，即辛指标间隙（symplectic index gap）。它决定了给定格上是否存在正则的、良局部化的窗函数。例如，我们证明 Schwartz 类 Gabor 框架窗存在当且仅当该间隙至少等于信号空间维数，其中对辛无理格采用扩展间隙约定。我们提出了辛指标间隙 Balian-Low 定理，将经典 Balian-Low 定理和融合 Balian-Low 定理置于调制空间尺度下的统一框架中。对于辛有理格——包括由有理向量张成的格，也是应用中最重要的情形——调制空间正则性的必要条件也是充分的，除了剩余的低间隙无穷远端点之外。配套论文《Tilings, Packings, and Smooth and Compactly Supported Gabor Windows》证明了相关的几何 Tiling-ε-Packing 定理，并构造了光滑且紧支撑的 Gabor 窗。
英文摘要
A lattice Gabor frame consists of a window function and its time-frequency shifts along a lattice. We classify the lattices that admit Schwartz-class and, equivalently, Feichtinger frame windows. Except for symplectically irrational lattices of critical covolume 1,, we determine the lattice-dependent achievable window smoothness and decay using the scale of modulation spaces. For symplectically rational lattices, including the rational lattices typically used in digital applications, the classification is governed by an integer lattice parameter introduced herein, the symplectic index gap. Our results extend the classical and amalgam Balian-Low theorems. The companion paper "Tilings, Packings, and Smooth and Compactly Supported Gabor Windows" develops the associated tiling-packing theorem and compact-support constructions.
Comments108 pages, 12 figures
